Problem: Given $ m \angle ABC = 5x + 3$, $ m \angle CBD = 6x - 75$, and $ m \angle ABD = 93$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Solution: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {5x + 3} + {6x - 75} = {93}$ Combine like terms: $ 11x - 72 = 93$ Add $72$ to both sides: $ 11x = 165$ Divide both sides by $11$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 6({15}) - 75$ Simplify: $ {m\angle CBD = 90 - 75}$ So ${m\angle CBD = 15}$.